I'm travelling to Tashkent, Uzbekistan in mid-June. What are the transport options going to Kyrgyzstan/Kazakhstan and back regarding time/cost/ease of arranging? Specifically I'm interested in travelling the Tashkent - Osh - Arslanbob - Song Kul - Bishkek - Shymkent - Tashkent circle.
I've been to all those places except for Shymkent last summer (2014). But I did not follow your suggested route and walked and hitched a lot but also used some public transport options.
Even though there are some bigger buses on some of the more popular routes, you will mostly rely on small buses and shared taxis.
Even between Osh and Bishkek, the two biggest cities in Kyrgyzstan there are no regular bus services. Most people take a shared taxi or fly.
On the other hand, the main bus station in Bishkek has lots of regular buses to Almaty and Shymkent in Kazakhstan and many other places.
Arslanbob is a small place in the hills, you wont find a bus from Osh and even finding a shared taxi will be difficult. You can take a private taxi. You can also take a shared taxi from Osh to Jalal-Abad and then another one or a mini-bus to Arslanbob. In Arslanbob don't expect to find a taxi to Bishkek or Song Kul, take a mini bus back to Jalal-Abad.
I'm not sure how easy it is to reach Song Kul from the south, most people come from Bishkek. To get to the lake itself, there is no public transport, your have to arrange a tourist bus or taxi in Kochkor. Many people get on horse.
In general you find out where the buses/taxis are leaving from and go there in the morning. For shared taxis you always have to hackle about the price, this works even if you don't speak Russian. You never book anything in advance. The car or SUV leaves when its full, unless you pay for all seats (which can be more than you think).
Tashkent has some overland buses, but even a popular route like the one to Samarkand is mostly done by shared taxi.
I don't know about the route from Tashkent to Osh, as I traveled through Tajikistan.
So traveling in that region is rather adventurous, knowing some Russian helps a lot but I managed without.
Unless you want to go to a really remote area, there are ways to get there, that's how the locals travel.
The main roads in Kyrgyzstan are actually in a pretty good condition, but because it is a very mountainous country, travel takes time.
I can't give any specific costs and time estimates, but it's all reasonable. Remember, this is how the locals travel and they are much poorer than you and me.